How do you prove that a function is differentiable at a given point?

if you want to check if f(x) has a derivative at a

then

you need to calculate the following limit:

lim h->0 ( f(a+h)-f(a) ) / h

`

Okay2f43b42fd833d1e77420a8dae7419002f43b42fd833d1e77420a8dae7419002f43b42fd833d1e77420a8dae741900 you recognize that it is not non-end at x = a million/22f43b42fd833d1e77420a8dae741900 So what might take place if it have been differentiable at x = a million/2? Then the thought you reported as [2f43b42fd833d1e77420a8dae7419002f43b42fd833d1e77420a8dae7419002f43b42fd833d1e77420a8dae741900] that if a function is differentiable, it additionally must be non-end might propose that that's non-end at x = a million/22f43b42fd833d1e77420a8dae741900 yet that's ridiculous, because we already comprehend that it is not non-end at x = a million/22f43b42fd833d1e77420a8dae741900 for this reason it can't be differentiable at x = a million/22f43b42fd833d1e77420a8dae741900 ---------- That theorem has yet another variety, hinted by potential of the above: If a function isn't non-end at a ingredient, then it is not differentiable at that point2f43b42fd833d1e77420a8dae741900

Show that the function is continuous at that point (doesn't have a hole or asymptote or something) and that the limit as x (or whatever variable) approaches that point from all sides is the same as the value of the function at that point.

To prove a function is differentiable at point p:

lim(x->p-) = lim(x->p+) = f(p)

A function is differentiable if the limit of the difference quotient exists:

lim h --> 0 [ ( f(x + h) - f(x) ) / h ]

for a function f(x) to be differentiable at a

1.f(a) mustexist

2.limit x>a shouldexist

3.limit x>a must beequal to f(a)

there are value from left and right are the same